ON CONSTRUCTION OF ALMOST-RAMANUJAN GRAPHS

O. Reingold et al. introduced the notion zig-zag product on two different graphs, and presented a fully explicit construction of d-regular expanders with the second largest eigenvalue O(d-1/3). In the same paper, they ask whether or not the similar technique can be used to construct expanders with the second largest eigenvalue O(d-1/2). Such graphs are called Ramanujan graphs. Recently, zig-zag product has been generalized by A. Ben-Aroya and A. Ta-Shma. Using this technique, they present a family of expanders with the second largest eigenvalue d-1/2 + o(1), what they call almost-Ramanujan graphs. However, their construction relies on local invertible functions and the dependence between the big graph and several small graphs, which makes the construction more complicated. In this paper, we shall give a generalized theorem of zig-zag product. Specifically, the zig-zag product of one "big" graph and several "small" graphs with the same size will be formalized. By choosing the big graph and several small graphs individually, we shall present a family of fully explicitly almost-Ramanujan graphs with locally invertible function waived.

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NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972710001644 ◽

2010 ◽

Vol 83 (1) ◽

pp. 87-95

Author(s):

KA HIN LEUNG ◽

VINH NGUYEN ◽

WASIN SO

Keyword(s):

Regular Graph ◽

Elementary Proof ◽

Explicit Construction ◽

Simple Graph ◽

Regular Graphs ◽

Circulant Graph ◽

Circulant Graphs ◽

Largest Eigenvalue ◽

Image Position ◽

Second Largest Eigenvalue

AbstractThe expansion constant of a simple graph G of order n is defined as where $E(S, \overline {S})$ denotes the set of edges in G between the vertex subset S and its complement $\overline {S}$. An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.

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Graphs with the second signless Laplacian eigenvalue ≤ 4

Special Matrices ◽

10.1515/spma-2021-0152 ◽

2021 ◽

Vol 10 (1) ◽

pp. 131-152

Author(s):

Stephen Drury

Keyword(s):

General Solution ◽

Knapsack Problem ◽

Laplacian Eigenvalue ◽

Simple Graphs ◽

Largest Eigenvalue ◽

Signless Laplacian ◽

Second Largest Eigenvalue

Abstract We discuss the question of classifying the connected simple graphs H for which the second largest eigenvalue of the signless Laplacian Q(H) is ≤ 4. We discover that the question is inextricable linked to a knapsack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.

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